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Does category theory deny them like Skolem did
one of these things is not like the other. Category theory is not a person who can take a philosophical position. A better comparison would be to compare catgory theory with first-order logic, or ZFC, or Peano Arithmetic. Category theory per se is a framework, a language, a foundation, depending on your point of view. Vanilla category theory says nothing about numbers a priori, it’s about objects and morphisms, functors, natural transformations and so on. If one gets to the point of defining real numbers in some kind of category-theoretic foundation or framework, then one might as well be working in some other foundation, since all these things are interconvertible. ETCS+R says the same things about real numbers as ZFC. Real numbers in a more general topos behave the same as in a constructive foundation like IZF.
Is Chaitins constant which is frequently given as an example of an undefinable number really even a number?
It’s perfectly well-defined in the standard definition of real numbers, yes. I don’t know about how it works in constructive logic.
numbers dont really even exist
This is beyond the scope of mathematics. There are plenty of mathematicians that are not Platonists, or even anti-Platonists, and there are plenty of mathematicians who are strongly Platonist, even happy to say that arbitrarily large infinite sets “exist”. One can be agnostic on this and just ask what theorems follow from what axioms, and this no one disagrees on, once the proof is pinned down sufficiently.
But, on the flip side, you have someone like Ramanujan
Ramanujan was talking about integers, IIRC, not real numbers. When you say “numbers” you need to be more specific.
Secondly, could you please enlighten me if you would on what undefinable number means in standard set theory?
Set theory is a red herring here. It’s about having a formal language and that we only write down finitely many symbols drawn from a countable pool. Even allowing for things like non-halting Turing machines that calculate successive, improving, approximations to a real number, or convergent infinite sums and so on, these are still finite descriptions.
“Definability” in set theory also applies to something else that is not about real numbers at all, and involves transfinite recursion.
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